Minkowski space
Let us consider a photon moving with the speed of lightc. It shall propagate for a distanced~xin an infinitesimal timedt, i.e.,
c2dt2−d~x2= 0. (1.1)
If we consider a transformation of space and time coordinates(t, x)to(t0, x0)the statement that the speed of lightcis the same in the new coordinate system is equivalent to the statement that also
c2dt02−d~x02 = 0. (1.2)
This property can now be expressed in a convenient mathematical representation by introducing vectors in a four-dimensional pseudo-Euclidean vector space with metric
(gµν) =
1
−1
−1
−1
. (1.3)
This vector space is calledMinkowski space. The infinitesimal timedtand the corresponding vector d~xare combined to a four vector(dxµ) = (c dt, d~x)and Eq. (1.1) can be written as
ds2=dxµdxµ=gµνdxµdxν = 0. (1.4)
In this framework the transformations of coordinates that leave the speed of light invariant are just the isometries that leave the inner products of infinitesimal difference vectors invariant.
For convenience we adopt natural units in the remainder of this thesis and setc=~= 1.
Poincar ´e group
The group of isometries of the Minkowski space is thePoincar´e groupconsisting of all transforma-tions of the affine form
x0µ= Λµνxν+Tµ, (1.5)
with
dx0µdx0µ=dxµdxµ (1.6)
wheredxµis an infinitesimal difference vector in Minkowski spacetime,Λµν is a real four-by-four matrix andTµis a four vector describing translations. Equation (1.6) implies that
dx0µdx0µ=gµνΛµαΛνβdxαdxβ =gαβdxαdxβ (1.7) or
gµνΛµαΛνβ =gαβ. (1.8)
This defining condition can be written in matrix form as
ΛTgΛ =g (1.9)
and thus
det Λ =±1. (1.10)
The subgroup defined byTµ= 0, i.e., the subgroup of all linear transformations, is theLorentz group and its elements are calledLorentz transformations.
Restricted Lorentz group
Let us first consider Lorentz transformations that are continuously connected to the identity transfor-mation Λ = 1. Lorentz transformations with this property live in a subgroup called therestricted Lorentz group. Since a continuous transformation cannot change the sign of Eq. (1.10), restricted Lorentz transformations havedet Λ = 1.
A well-known subgroup of the restricted Lorentz group is the group of rotations with
Λ =
1 0 0 0 0
0 R
0
, (1.11)
where RTR = 1, detR = 1. Now by first rotating the spatial components appropriately we can restrict the remaining discussion to the two-dimensional subspace of vectors(dxµ) = (dt, dx,0,0).
The relevant Lorentz transformations are then of the form
Λ =
Λ00 Λ01 0 0 Λ10 Λ11 0 0
0 0 0 0
0 0 0 0
. (1.12)
Thus for infinitesimal transformationsΛ = 1+Gthe defining condition of Eq. (1.9) yieldsGTg+ gG= 0, and therefore
0 = with arbitrarys∈R. Let us try to understand what the parametersmeans. Consider an infinitesimal vector(dt, dx)that transforms to
dt0
Now we define a transformed velocity v0 = dx0
dt0 = vcoshs+ sinhs
vsinhs+ coshs (1.16)
with v = dx/dt. If we have v = 0 in the untransformed system we have v0 = tanhs in the transformed system. Therefore transformations of this type describe a change of coordinates to a frame of reference that moves with a constant velocity of tanhs relative to the original frame of reference. These are theboostsin the special theory of relativity withrapiditys.
Let us defineβ = tanhs. Sincecosh2s−sinh2s= 1, we can show that coshs= 1
p1−tanh2s
= 1
p1−β2 =γ . (1.17)
Therefore we can express the transformation also by the matrix Λ(β) =
Consider the vector(xµ) = (t,0)which is invariant under rotations and transforms to (x0µ) =
tcoshs tsinhs
(1.19) under a boost with rapiditys. Sincecoshs > 0, we conclude that the sign ofx0 is invariant under boosts and thus under the complete restricted Lorentz group.
Therefore, in order to obtain all possible Lorentz transformations, the discrete Lorentz transforma-tion
needs to be included in addition to restricted Lorentz transformations. This is the time reversal operator. Furthermore thespace inversionorparityoperator
P =
1
−1
−1
−1
(1.21)
is also not a part of the restricted Lorentz group and needs to be included separately.
The quotient group of the Lorentz group and the restricted Lorentz group is the discrete group with elements
1, P, T, P T . (1.22)
In other words, the Lorentz group can be partitioned in four disconnected parts defined by
det Λ =±1, Sgn Λ00 =±1. (1.23) We call transformations with det Λ = 1 properLorentz transformations and transformations with Sgn Λ00 = 1orthochronousLorentz transformations.
Generators of the restricted Lorentz group
Recall that infinitesimal restricted Lorentz transformationsΛ =1+Gsatisfy
GTg+gG= 0. (1.24)
We writeGin block form
G=
G00 G01 G10 G11
, (1.25)
whereG00only acts on the temporal component,G11only acts on the spatial components, andG01 andG10mix spatial and temporal components. In this way Eq. (1.24) can be expressed as
0 =
1 0 0 −13
G00 G01
G10 G11
+
G00 GT10 GT01 GT11
1 0
0 −13
=
2G00 G01−GT10 GT01−G10 −G11−GT11
, (1.26)
where13 is the three-dimensional identity matrix. Therefore the defining conditions for generators of the restricted Lorentz group are
G01=GT10, G00= 0, GT11=−G11. (1.27) This implies the following generators of the restricted Lorentz group.
The boosts are generated by
Ki=
0 eTi ei 0
(1.28)
with(ei)j =δij andi= 1,2,3. They satisfy
withJkdefined below.
The rotations are generated by
Ji=
Hence boosts do not form a subgroup of the restricted Lorentz group, but rotations do. Note that [Ki, Jj] = The Lie algebra of the restricted Lorentz group is therefore given by
[Ki, Kj] =−εijkJk, [Ji, Jj] =εijkJk, [Ki, Jj] =εijkKk. (1.33) A finite transformation is given by
Λ = exp[~s·K~ +ϕ~·J~], (1.34) whereϕ~ contains the angles of a rotation and~scontains the rapidities of a boost.
A convenient representation of the generators is given by Si±= 1
2(±Ki+iJi) (1.35)
with(Si±)†=Si±andi= 1,2,3. We find
[Sia, Sjb] = (ab[Ki, Kj] +ib[Ji, Kj] +ia[Ki, Jj]−[Ji, Jj])/4
=iεijk[i[(1 +ab)/4]Jk+ [(a+b)/4]Kk] =δabiεijkSka. (1.36) Therefore the group algebra factorizes in a direct product of twoSU(2)algebras (this is of course not true in terms of groups). We can expressJiandKiin terms ofSi±as
iJi=Si++Si−, Ki =Si+−Si−. (1.37) Therefore Eq. (1.34) can be written as
Λ = exp[si(Si+−Si−)−iϕi(Si++Si−)] = exp[−ixiSi+] exp[−ix∗iS−i ] withxi=ϕi+isi.
Translations in space and time
The Casimir operators ofS+andS−can now be used to classify the representations of the restricted Lorentz group. These Casimir operators are, however, no invariants of representations of the complete Poincar´e group since they do not commute with all translations of space and time. In this section we show that the spin of a massive particle is, nevertheless, a well-defined quantity.
We extend the Minkowski space by a fifth dimension so that we can express a general transforma-tion of the Poincar´e group, see Eq. (1.5), conveniently as
x0 = Γ(Λ, T)x (1.38)
with Lorentz transformationΛ, a four-dimensional translation vector(Tµ),(xµ) = (x0, x1, x2, x3,1), and
Γ(Λ, T) =
Λ (Tµ)
0 1
(1.39) in block notation. The generators of translations in space and time Pµ are therefore given by the matrices
Pµ=
0 (δµν)
0 0
(1.40) in block notation. A finite translation is given by
Γ(1, T) = exp
3
X
µ=0
TµPµ
. (1.41)
We can now determine the algebra of the complete Poincar´e group,
[Pµ, Pν] = 0, [P0, Ji] = 0, [P0, Ki] =−Pi, [Pi, Jj] =εijkPk, [Pi, Kj] =−δijP0, [Ki, Kj] =−εijkJk,
[Ji, Jj] =εijkJk, [Ki, Jj] =εijkKk. (1.42) The Poincar´e algebra has two Casimir operators. The first one is given by
C1=PµPµ=P02−Pi2. (1.43)
We check explicitly that
[Pµ, C1] = 0, (1.44)
[Ji, C1] = [Ji, P02]−[Ji, Pj2] =−[Ji, Pj]Pj−Pj[Ji, Pj]
= 2εijkPkPj =−2εijkPkPj = 0, (1.45) [Ki, C1] = [Ki, P0]P0+P0[Ki, P0]−[Ki, Pj]Pj−Pj[Ki, Pj]
= 2PiP0−2PiP0 = 0 (1.46)
for arbitraryiandµ. Let us pause at this point and ask what this means for a theory of a free particle with energyE and momentum~p. In quantum mechanics the generator of the translations in space, Pi, measures theith component of the momentum, and the generator of the translations in time,P0,
measures the energy. Therefore if we letC1 act on a free particle state|E, ~pi with energyE and momentum~pwe find
C1|E, ~pi= (E2−~p2)|E, ~pi=m2|E, ~pi , (1.47) wheremis the mass of the particle. We can conclude that the mass of a particle is invariant under the Poincar´e group and can be considered a well-defined property of a particle.
The second Casimir operatorC2can be conveniently defined in terms of thePauli-Lubanski vector Wµwith
W0 =JjPj, Wi=P0Ji−εijkKjPk. (1.48) It is given by
C2=WµWµ= (W0)2−(Wi)2. (1.49) In order to prove thatC2is indeed a Casimir operator we first show thatWµcommutes with transla-tions, i.e.,
[Pµ, W0] = [Pµ, JjPj] = [Pµ, Jj]Pj = (1−δµ0)εµjkPkPj = 0, (1.50) [Pj, Wi] =P0[Pj, Ji]−εilk[Pj, Kl]Pk=P0Pk(εjik+εijk) = 0, (1.51) [P0, Wi] =−εilk[P0, Kl]Pk=εilkPlPk = 0. (1.52) Next we discuss the commutators ofWµwith boosts and calculate
[Kj, W0] = [Kj, JiPi] = [Kj, Ji]Pi+Ji[Kj, Pi]
=εjikKkPi+JjP0 =Wj (1.53)
and
[Kj, Wi] = [Kj, P0Ji]−εilk[Kj, KlPk]
=P0[Kj, Ji] + [Kj, P0]Ji−εilkKl[Kj, Pk]−εilk[Kj, Kl]Pk
=εjikP0Kk+PjJi−εiljKlP0+εilkεjlrJrPk
=εjik[P0, Kk] +PjJi+ (δijδkr−δirδkj)JrPk
=−εjikPk+ [Pj, Ji] +δijJkPk=−εjikPk+εjikPk+δijJkPk
=δijW0. (1.54)
We finally calculate the commutators ofWµwith rotations and find
[Jj, W0] = [Jj, JiPi] = [Jj, Ji]Pi+Ji[Jj, Pi] =εjikJkPi−εijkJiPk
=εjikJkPi−εkjiJkPi= 0, (1.55)
[Jj, Wi] = [Jj, P0Ji]−εilk[Jj, KlPk]
=εjikP0Jk−εilk[Jj, Kl]Pk−εilkKl[Jj, Pk]
=εjikP0Jk+εilkεljrKrPk+εilkεkjrKlPr
=εjikP0Jk+ (εilkεljr+εljkεirl)KrPk
=εjikP0Jk+ (δkjδir−δjrδik)KrPk
=εjikP0Jk+εljiεlkrKrPk
=εjik(P0Jk+εklrKrPl) =εjikWk. (1.56)
We observe thatWµ has the same commutation relations with the other parts of the algebra asPµ, and thereforeC2is also a Casimir operator.
For a massive particle we can calculate the action ofC2in its rest frame, i.e.,
C2|m,0i=−m2Ji2|m,0i . (1.57)
Therefore|m,0imust also be an eigenstate ofJi2and the corresponding eigenvaluess(s+ 1) corre-spond to thespinor intrinsic rotation of the point-like particle. In other words, massive particles can be classified according to their spin as defined by their behavior under the rotation group.
For a massless particle there is no rest frame and thus the situation is more complicated. It turns out that for massless particles the projection of the spin to the momentum,
λ=J~·P ,ˆ (1.58)
is a well-defined property and assumes the role of the spin of massive particles. This property is calledhelicity.
For a detailed discussion of the representation theory of the complete Poincar´e group we refer to Refs. [1, 3, 5].